3.2623 \(\int \frac{x^{-1-2 n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=78 \[ -\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{a^3 n \left (a+b x^n\right )}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]

[Out]

-1/(2*a^2*n*x^(2*n)) + (2*b)/(a^3*n*x^n) + b^2/(a^3*n*(a + b*x^n)) + (3*b^2*Log[
x])/a^4 - (3*b^2*Log[a + b*x^n])/(a^4*n)

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Rubi [A]  time = 0.115309, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{a^3 n \left (a+b x^n\right )}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - 2*n)/(a + b*x^n)^2,x]

[Out]

-1/(2*a^2*n*x^(2*n)) + (2*b)/(a^3*n*x^n) + b^2/(a^3*n*(a + b*x^n)) + (3*b^2*Log[
x])/a^4 - (3*b^2*Log[a + b*x^n])/(a^4*n)

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Rubi in Sympy [A]  time = 17.73, size = 73, normalized size = 0.94 \[ - \frac{x^{- 2 n}}{2 a^{2} n} + \frac{b^{2}}{a^{3} n \left (a + b x^{n}\right )} + \frac{2 b x^{- n}}{a^{3} n} + \frac{3 b^{2} \log{\left (x^{n} \right )}}{a^{4} n} - \frac{3 b^{2} \log{\left (a + b x^{n} \right )}}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1-2*n)/(a+b*x**n)**2,x)

[Out]

-x**(-2*n)/(2*a**2*n) + b**2/(a**3*n*(a + b*x**n)) + 2*b*x**(-n)/(a**3*n) + 3*b*
*2*log(x**n)/(a**4*n) - 3*b**2*log(a + b*x**n)/(a**4*n)

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Mathematica [A]  time = 0.160889, size = 62, normalized size = 0.79 \[ -\frac{x^{-2 n} \left (a^2+\frac{2 b^3 x^{3 n}}{a+b x^n}-4 a b x^n\right )+6 b^2 \log \left (a x^{-n}+b\right )}{2 a^4 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 - 2*n)/(a + b*x^n)^2,x]

[Out]

-((a^2 - 4*a*b*x^n + (2*b^3*x^(3*n))/(a + b*x^n))/x^(2*n) + 6*b^2*Log[b + a/x^n]
)/(2*a^4*n)

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Maple [A]  time = 0.039, size = 117, normalized size = 1.5 \[{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( -3\,{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}n}}-{\frac{1}{2\,an}}+{\frac{3\,b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }-3\,{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-2*n)/(a+b*x^n)^2,x)

[Out]

(-3*b^3/a^4/n*exp(n*ln(x))^3-1/2/a/n+3/2*b/a^2/n*exp(n*ln(x))+3*b^2/a^3*ln(x)*ex
p(n*ln(x))^2+3*b^3/a^4*ln(x)*exp(n*ln(x))^3)/exp(n*ln(x))^2/(a+b*exp(n*ln(x)))-3
*b^2/a^4/n*ln(a+b*exp(n*ln(x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^2,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.232531, size = 142, normalized size = 1.82 \[ \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, a^{2} b x^{n} - a^{3} + 6 \,{\left (a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} - 6 \,{\left (b^{3} x^{3 \, n} + a b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b n x^{3 \, n} + a^{5} n x^{2 \, n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^2,x, algorithm="fricas")

[Out]

1/2*(6*b^3*n*x^(3*n)*log(x) + 3*a^2*b*x^n - a^3 + 6*(a*b^2*n*log(x) + a*b^2)*x^(
2*n) - 6*(b^3*x^(3*n) + a*b^2*x^(2*n))*log(b*x^n + a))/(a^4*b*n*x^(3*n) + a^5*n*
x^(2*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1-2*n)/(a+b*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-2 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(-2*n - 1)/(b*x^n + a)^2,x, algorithm="giac")

[Out]

integrate(x^(-2*n - 1)/(b*x^n + a)^2, x)